The Conservative MP did not deny the accusations about events in 2010, made by his former assistant Caroline Edmondson in the Mail on Sunday.

“I’m not going to be dishonest,” Garnier said. He insisted that referring to Edmondson as “sugar tits”, as she says he did, was a reference to the popular BBC comedy Gavin and Stacey, saying: “It absolutely does not constitute harassment.”

The Cabinet Office inquiry was announced by Jeremy Hunt, the health secretary, during an interview on BBC1’s The Andrew Marr Show on Sunday.

He said: “The stories, if they are true, are obviously totally unacceptable and the Cabinet Office will be conducting an investigation into whether there’s been a breach of the ministerial code in this particular case.”

Theresa May later wrote to the Speaker of the House of Commons, John Bercow, saying that the Commons disciplinary procedures lacked “teeth” and required urgent reform.

She said a situation where MPs did not have to follow procedures laid down by the Independent Parliamentary Standards Authority (Ipsa) could not be tolerated any longer.

She asked for Bercow’s assistance in working on a cross-party basis to establish a new “house-wide mediation service” backed by a “contractually binding grievance procedure” available for all MPs.

A Conservative spokesman said: “As the prime minister has made clear, any reports of sexual harassment are deeply concerning and any unwanted sexual behaviour is completely unacceptable in any walk of life, including politics.

“Today, as a result of allegations about a serving minister, the prime minister has also asked the Cabinet Office to conduct an immediate investigation to see whether those reported actions break the ministerial code.”

According to the newspaper report, Garnier gave Edmondson money to buy a sex toy for his wife and another for a woman working in his Wyre Forest constituency office.

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Edmondson said: “He suggested to me in a Commons bar one evening that we went shopping for sex toys in Soho. The next day, he said: ‘Come on, let’s do it.’ He took me to Soho and gave me the money to buy two vibrators. He stood outside the shop while I did.”

Garnier, a former City fund manager, is married with three children and was elected MP for Wyre Forest in 2010.

Downing Street has not yet responded to Garnier’s case, but a separate report in the Sunday Times claimed May receives regular briefings on the sexual affairs of members of the government from Conservative whips, who are responsible for enforcing party discipline.

Labour is also braced for fresh allegations of inappropriate behaviour, as junior staff feel emboldened to come forward.

Diane Abbott, the shadow home secretary, told Marr: “This isn’t a problem for any particular party or faction of a party ... it runs right across parliament.” She said she hoped the recent revelations would lead to a sea change in the culture of all male-dominated professions.

Some MPs have claimed allegations are not taken seriously enough. One former Conservative minister said: “It’s grim: the whole culture has to change.”

Another senior Tory named on Sunday was Stephen Crabb, who stood for the party leadership last year. He has admitted sending a 19-year-old who hoped to work for him a series of messages that he described as “sexual chatter”.

“We exchanged messages which talked about sex but none of it was meant seriously,” Crabb told the Sunday Telegraph. “We met for coffee a few times and had a glass of wine once at the Commons, but nothing more. I accept any kind of sexual chatter like this is totally wrong and I am sorry for my actions,.” The incident is separate from one last year in which the MP sent explicit text messages to another young woman.

The Weinstein affair, in which the former Hollywood mogul has been accused of multiple incidents of rape and sexual harassment, many of them by women who hoped he would further their careers, has raised questions about the culture at Westminster.

Michael Gove, the environment secretary, apologised on Saturday after making a joke about the Hollywood producer on BBC Radio 4’s Today programme.

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Gove makes 'clumsy' Harvey Weinstein joke on Today programme – audio

He told the presenter John Humphrys during an interview to mark the show’s 60th anniversary: “Sometimes I think coming into the studio with you, John, is a bit like going into Harvey Weinstein’s bedroom: you hope to emerge with your dignity intact.”

Abbott told Marr on Sunday: “Particularly in parliament, making sexual harassment a joke is one of the reasons it’s not been dealt with. You have got to realise it’s undermining and demeaning for women and it undermines and demeans the institution.”